Kuratowski closure axioms

Encyclopedia : K : KU : KUR : Kuratowski closure axioms

---

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first introduced by Kazimierz Kuratowski, in a slightly different form that applied only to Hausdorff spaces.

A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.

Definition

A topological space [(X,cl)] is a set [X] with a function

[cl:\mathcal(X) \to \mathcal(X)]

called the closure operator where [\mathcal(X)] is the power set of [X].

The closure operator has to satisfy the following properties

1. [ A \subseteq cl(A) \! ] (Extensivity)
2. [ cl(cl(A)) = cl(A) \! ] (Idempotence)
3. [ cl(A \cup B) = cl(A) \cup cl(B) \! ] (Preservation of binary unions)
4. [ cl(\varnothing) = \varnothing \! ] (Preservation of nullary unions)

Notes

Axioms (3) and (4) can be generalised (using a proof by mathematical induction) to the equivalent single statement:

[ cl(A_ \cup \cdots \cup A_) = cl(A_) \cup \cdots \cup cl(A_), n \geq 0 \! ] (Preservation of finitary unions).

An operator that only satisfies axioms (1) and (2) is called a Moore closure. Moore closure operators are often studied in lattice theory.

Recovering topological definitions

A function between two topological spaces

[f:(X,cl) \to (X',cl')]

is called continuous if for all subsets [A] of [X]

[f(cl(A)) \subset cl'(f(A))]

A point [p] is called close to [A] in [(X,cl)] if [p\in cl(A)]

[A] is called closed in [(X,cl)] if [A=cl(A)]. In other words the closed sets of [X] are the fixed points of the closure operator.

 

---

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.

Search Titles

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

?